Deep Fuzzy Credibility Surfaces for Integrating External Databases in the Estimation of Operational Value at Risk

Operational risk (OR) is usually caused by losses due to human errors, inadequate or defective internal processes, system failures, or external events that affect an organization. According to the Basel II agreement, OR is defined by seven risk events: internal fraud, external fraud, labor relations, clients, damage to fixed assets, technical failures and failures in the execution and administration of processes. However, the low frequency with which a loss event occurs creates a technological challenge for insurers in estimating the operational value at risk (OpVar) for the protection derived from an organization’s business activities. Following the above, this paper develops and analyzes a Deep Fuzzy Credibility Surface model (DFCS), which allows the integration in a single structure of different loss event databases for the estimation of an operational value at risk (OpVar), overcoming the limitations imposed by the low frequency with which a risk event occurs within an organization (sparse data). For the estimation of OpVar, the DFCS model incorporates a novel activation function based on the generalized log-logistic function to model random variables of frequency and severity that define a loss event (linguistic random variables), as well as a credibility surface to integrate the magnitude and heterogeneity of losses in a single structure as a result of the integration of databases. The stability provided by the DFCS model could be evidenced through the structure exhibited by the aggregate loss distributions (ALDs), which are obtained as a result of the convolution process between frequency and severity random variables for each database and which are expected to achieve similar structures to the probability distributions suggested by Basel II agreements (lean, long tail, positive skewness) against the OR modeling. These features make the DFCS model a reference for estimating the OpVar to protect the risk arising from an organization’s business operations by integrating internal and external loss event databases.